Spherical Cubic Blends: $\mathcal{C}^{2}$-Continuous, Zero-Clamped, and Time-Optimized Interpolation of Quaternions

Modern collaborative robotic applications require robot motions that are predictable for human coworkers. Therefore, trajectories often need to be planned in task space rather than configuration space ($\mathcal{C}$-space). While the interpolation of translations in Euclidean space is straightforward, the interpolation of rotations in $SO(3)$ is more complex. Most approaches originating from computer graphics do not exhibit the often desired $\mathcal{C}^{2}$-continuity in robotics. Our main contribution is a $\mathcal{C}^{2}$-continuous, zero-clamped interpolation scheme for quaternions that computes a fast synchronized motion given a set of waypoints. As a second contribution, we present modifications to two state-of-the-art quaternion interpolation schemes, Spherical Quadrangle Interpolation (SQUAD) and Spherical Parabolic Blends (SPB), to enable them to compute $\mathcal{C}^{2}$-continuous, zero-clamped trajectories. In experiments, we demonstrate that for the time optimization of trajectories, our approach is computationally efficient and at the same time computes smooth trajectory profiles.